Local Semicircle Law in the Bulk for Gaussian β-Ensemble

We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of n−1+δ for any δ>0. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of n−1/2+δ; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erdös, and Yau in arXiv:1104.2272 [math.PR] and Bao and Su in arXiv:1104.3431 [math.PR]. Our approach is new and could be extended to other tridiagonal models.